In this manuscript, Section  introduces notation for GSDs with stopping rules dependent on a parameter of interest [through the distributions of the test statistics]. This section also presents distributions of cumulative test statistics conditional on reaching a stage, conditional on stopping at a stage, and unconditional defined on the probability space with truncation-adapted support. All of these probability distributions are truncated or truncated-mixtures. Section  presents likelihood-based inference on the truncation-adapted probability space. In section REF  a local asymptotic distribution of the MLEs is found to be non-degenerate; it is a mixture of truncated normal distributions. Section REF  shows that the possibility of early stopping does not change the monotonicity of likelihood ratios in the one-parameter exponential family. Thus, stage-specific tests continue to be uniformly most powerful by the Karlin-Rubin theorem, which makes sequential tests based on monotone likelihood ratio uniformly most powerful. Throughout Sections  and  theoretical results are illustrated by a two stage example, [1]}. Finally, Section  concludes this article with a summary and a discussion of impact within the contemporary research environment.
